During the third and final presidential debate Monday night, President Obama and Mitt Romney disputed an array of statements on foreign policy. Here are a few worth a deeper look:

Defense spending

Claim: Obama said Romney wants to add $2 trillion in spending the military hasn't asked for and that defense spending has increased every year he has been president.

The facts: Obama's claim about Romney's increase is accurate; his statement that budgets have increased is not.

Romney calls for spending a minimum of 4% of the nation's gross domestic product on defense. Over 10 years, that would amount to about $2 trillion more for the Pentagon than Obama has budgeted over the same period.

The 2013 Pentagon base budget - excluding costs for the wars in Afghanistan and Iraq - as proposed by Obama declines by $5 billion compared with 2012, according to the Pentagon's comptroller.

Syria

Claim: Obama said that Romney said he would provide heavy arms to Syrian rebels.

The facts: Romney did say he would provide heavy weaponry to rebels in Syria. In an Oct. 8 speech in Lexington, Va., Romney said he "will work with our partners to identify and organize those members of the opposition who share our values and ensure they obtain the arms they need to defeat (President Bashar) Assad's tanks, helicopters and fighter jets."

The 'apology tour'

Claim: Romney said Obama went on "an apology tour of going to various nations in the Middle East and criticizing America."

The facts: The use of the term "apology tour" to describe Obama's April 2009 foreign visits appears to have started with Fox News host Sean Hannity.

In his book, Romney cited a number of speeches in which he said Obama apologized: "The United States certainly shares blame" for the global banking meltdown, Obama told the French. The George W. Bush administration had "lowered our standing in the world," he told the English. And to the Turkish parliament, he said: "The United States is still working through some of our own darker periods in our history," which included slavery, segregation and treatment of American Indians.

Even so, none of his foreign speeches included what many people would consider an essential element of an apology: the words "we're sorry." That's why Obama is correct that professional fact-checkers have rated the statement as untrue.

Mali

Claim: Romney, citing a litany of Middle East hotspots, said northern Mali "has been taken over by al-Qaeda-type individuals."

The facts: Mali, an African nation of 14 million people in the western Sahara desert, has been embroiled in conflict this year as insurgent groups have fought for independence. The Economic Community of West African States has identified at least three of the groups as having links with al-Qaeda. Intelligence officials say the groups may also have connections to insurgent groups in Algeria and Libya.

The Obama administration's response has been low-key, but on Monday, a French defense official told the Associated Press that it was discussing drone strikes with the United States. German Chancellor Angela Merkel also sounded the alarm Monday, telling a German military conference near Berlin that "Free democratic states cannot accept international terrorism gaining a safe refuge in the north of the country."

Libya

Claim: Obama said Romney suggested that getting rid of Moammar Gadhafi in Libya was "mission creep."

The facts: "What we are watching in real time is another example of mission creep and mission muddle," Romney wrote in the National Review on April 21, in the midst of the Libya operation. In that piece, Romney said he supported the "specific, limited mission" of a no-fly zone to protect Libyan civilians from the Gadhafi regime, but he said Obama owed Americans a better explanation of why he had changed his position to call for the Libyan dictator's ouster.

After Gadhafi was killed by rebel forces, Romney said, "The world is a better place with Gadhafi gone."

The facts: When the U.S. government was trying to secure a status of forces agreement last year with the Iraqi government that would have allowed some U.S. troops to remain in the country, Romney said more U.S. troops should remain than Obama was proposing.

Romney repeated that sentiment in a video leaked to Mother Jones from a May fundraiser. Romney said: "This president's failure to put in place a status of forces agreement allowing ten to 20,000 troops to stay in Iraq: unthinkable." But there is no record that Romney made the claim as recently as "a few weeks ago."

Global Influence

Claim: Romney said nowhere in the world is the United States' role greater than it was four years ago.

The facts: Global attitudes about the United States have declined slightly over the past four years, according to the Pew Research Center. In 2008, 84% to 14% positive-negative view of the United States and 14% unfavorable. In 2012, that favorability figure had fallen to 80%-14%.

Veterans

Claim: Obama said the unemployment rate for veterans is below the national jobless rate.

The facts: The unemployment rate for veterans in September was 6.7%, just above the three-year low of 6.6% reached the previous month, according to the Bureau of Labor Statistics and the Department of Veterans Affairs. That's below the 7.8% national jobless rate. However, unemployment for veterans of the Iraq and Afghanistan era was 9.7% last month.

Iran

Claim: Obama said Romney now favors bilateral talks with Iran, a reversal of his previous stance.

The facts: The Associated Press reported that Romney refused to answer when asked Sunday whether he supported one-on-one talks.

Russia

Claim: Obama said Romney has called Russia the largest geopolitical threat to the United States.

The facts: It is true that Romney said in a March 26 interview with CNN that Russia "is without question our No. 1 geopolitical foe." He added: "They fight for every cause for the world's worse actors. The idea that he (Obama) has more flexibility in mind for Russia is very, very troubling indeed." In a subsequent interview, Romney sought to clarify his remarks, saying, "That doesn't make them an enemy. It doesn't make them a combatant. They don't represent the No. 1 national security threat."

Class size

Claim: Obama said Romney said that reduced class sizes at schools did not make a difference.

The facts: During a May visit to a Philadelphia charter school, Romney discussed his approach to education when he was governor of Massachusetts. He said people told him that smaller class sizes were needed, and Romney said he analyzed every school district and their students' performance.

"I said let's compare the average classroom size from each school district with the performance of our students, because we test our kids, and we'll see if there's a relationship. And there was not."

He later cited a study by a consulting firm that studied classroom size around the world and came to a similar conclusion. "So it's not the classroom size that's driving the success of those school systems."

Oil imports

Claim: Obama said the United States has cut its oil imports to the lowest levels in 20 years.

The facts: That's close. The Department of Energy said this year that U.S. dependence on imported oil fell to 45% last year, the first time it dropped below 50% since 1997. The White House, citing DOE figures, says on its website that net imports - that's imports minus exports - as a share of total consumption fell to 45% last year and that was the lowest level in 16 years. Meanwhile, U.S. oil production has risen sharply. September's domestic production was the highest of any September since 1998, the American Petroleum Institute reported last week. Overall, petroleum imports fell to 10.5 million barrels a day in September, down 602,000 barrels a day from a year earlier.

Cooperation with Israel

Claim: Obama said, "We have created the strongest military and intelligence cooperation between our two countries in history."

The facts: By its nature, intelligence cooperation is difficult to verify. But military cooperation is a bit more visible. On Sunday, the United States and Israel launched a joint training operation called "Austere Challenge 12." Assistant U.S. Secretary of State Andrew Shapiro called the exercise the "largest and most significant joint exercise in the allies' history."

Nevertheless, the event also happens at a low point in the political relationship between Obama and Israeli Prime Minister Benjamin Netanyahu. Obama declined to meet with Netanyahu last month when Israeli leader visited the U.S. And asked by Steve Kroft of 60 Minutes whether he felt pressure on Iran from Israel, Obama said he would "block out any noise" from the Israelis.

Middle East peace talks

Claim: Romney said the United States has not exerted leadership in the Middle East and said Israel and the Palestinians haven't met in two years

The facts: The last direct talks between Israeli and Palestinian leaders happened in September 2010 - and then quickly broke down when the Israeli moratorium on settlements in the West Bank expired.

Obama himself has admitted that peace in the Middle East has eluded him, as it has every president since Jimmy Carter. "I have not been able to move the peace process forward in the Middle East the way I wanted. It's something we focused on very early. But the truth of the matter is that the parties, they've got to want it as well," he told Washington, D.C., television station WJLA in July.

Romney is more pessimistic. In a secretly taped fundraiser more famous for his "47%" remark, he said: "The pathway to peace is almost unthinkable to accomplish," that "it's going to remain an unsolved problem" and that a permanent agreement is "just wishful thinking." He also angered many Palestinians in July when he said at a Jerusalem fundraiser that cultural differences accounted for the stark disparities in wealth between Israel and Palestine.

Romney has softened that stance more recently. In a foreign policy speech in Virginia this month, Romney said he would "will recommit America to the goal of a democratic, prosperous Palestinian state living side by side in peace and security with the Jewish state of Israel." By that score, Romney's Palestinian policy is the same as Obama's. They both espouse a "two-state" solution that first became U.S. policy under President George W. Bush.

Carter, with a delegation of elder statesmen visiting Israel Monday, said prospects of a lasting Israeli-Palestinian accord are "vanishing."

U.S.- Israel Relations

Claim: Romney said Obama was trying to create "daylight" between the United States and Israel.

The facts: The reported "daylight" policy first surfaced in a Washington Post story recounting a 2009 meeting Obama had with Jewish leaders, in which one leader told Obama, "If you want Israel to take risks, then its leaders must know that the United States is right next to them."

Obama disagreed, saying, "Look at the past eight years. ... During those eight years, there was no space between us and Israel, and what did we get from that? When there is no daylight, Israel just sits on the sidelines, and that erodes our credibility with the Arab states."

By that, the president seemed to be saying that the United States can be more effective acting as an honest broker between the Israelis and the Palestinians.

The White House hasn't disputed the Washington Post account, but press secretary Jay Carney has made it clear that the policy, if there is one, does not extend to Iran. "There is no daylight between the United States and Israel when it comes to the absolute need to prevent Iran from acquiring nuclear weapons," Carney said.

China

Claim: Obama said his administration has brought more trade cases against China in four years than the Bush administration did in eight years.

The facts: The Obama administration has brought eight trade cases against China with the World Trade Organization, according to PolitiFact.com and the Alliance for American Manufacturing. Bush filed seven cases over two terms. However, China joined the WTO in 2001, after President George W. Bush took office, and member countries effectively gave China a grace period. The United States was the first nation to file a trade case against China in 2004. Thus, Obama could take advantage of some of the groundwork laid by Bush's administration.

Exports to China

Claim: Obama said U.S. exports to China have doubled since the start of the Obama administration.

The facts: The United States has more than doubled its exports to China during Obama's presidency, but that has coincided with rising imports from China as well. When Obama was inaugurated in January 2009, the United States exported $4.2 billion worth of goods to China. By August 2012, that had risen to $8.6 billion, according to the U.S. Census Bureau. Imports from China over that time period have risen from $24.7 billion to $37.3 billion. So the American trade balance with China has gone from $20.6 billion in favor of China in January 2009 to $28.7 billion in August 2012.

Education

Claim: Romney and his running mate, Paul Ryan, want to cut the education budget, according to Obama.

The facts: Though Ryan's budget has called for $5.3 trillion cuts in federal spending over the next decade, he has not singled out education programs for reduction or elimination, according to Politifact. An analysis by the National Education Association determined that the large cuts proposed by Ryan, and generally embraced by Romney, could cut 2 million spots in the Head Start early childhood education program.

Romney insists that he is not going to cut education programs. "I'm not going to cut education funding. I don't have any plan to cut education funding and grants that go to people going to college. I'm planning on continuing to grow, so I'm not planning on making changes there," Romney said in his Oct. 3 debate with Obama.

The facts: Panetta and the Joint Chiefs of Staff have agreed to $487 billion in budget cuts. However, about $500 billion in additional cuts to defense spending will automatically begin in January - a process termed "sequestration" - if Congress and Obama do not reach a comprehensive deal to reduce deficits. Panetta was referring to those $500 billion in cuts as devastating. Moreover, Romney's running mate, Ryan, chairman of the House Budget Committee, voted for that Budget Control Act that included the automatic cuts.